Question 155
Determine the nonconstant polynomial functions $f:[0,1] \rightarrow \mathbb{R}^*$ with rational coefficients, which have the following property: for every $x$ in the interval $[0,1]$ there exist two polynomial functions $g_x, h_x:[0,1] \rightarrow \mathbb{R}$ with rational coefficients, such that $h_x(x) \neq 0$ and
$$
\int_{0}^{x} \frac{1}{f(t)} \mathrm{d} t=\frac{g_x(x)}{h_x(x)}
$$
$$
\int_{0}^{x} \frac{1}{f(t)} \mathrm{d} t=\frac{g_x(x)}{h_x(x)}
$$